基于改进复制动态演化博弈模型的最优防御策略选取

doi:10.11959/j.issn.1000-436x.2018010

摘要/Abstract

摘要：

针对同一博弈群体之间存在策略依存性，通过引入激励系数，改进传统复制动态方程，完善复制动态速率计算方法，构建基于改进复制动态的网络攻防演化博弈模型。利用改进复制动态方程进行演化均衡求解，采用雅可比矩阵的局部稳定分析法对所求均衡点进行稳定性分析，得到不同条件下的最优防御策略。研究结果表明，同一群体的不同策略之间既存在促进作用，也存在抑制作用。通过实验仿真验证了所提模型和方法的准确性和有效性，为解决现实社会中的信息安全问题提供了新的理论支撑。

关键词: 策略依赖性, 激励系数, 改进复制动态, 复制动态速率, 雅可比矩阵, 最优防御策略

Abstract:

In terms of the existence of strategy dependency in the same game group,network attack-defense evolutionary game model based on the improved replicator dynamics was constricted by introducing the intensity coefficient,which completed the method of calculating replicator dynamic rate.The improved replicator dynamic equation was adopted to solve the evolutionary equilibrium for the situation that both attack and defense have two optional strategies.The stability of the equilibrium points was analyzed by the local stability analysis method of Jacobian matrix,and the optimal defense strategies were obtained under different conditions.The results show that the strategy dependency between the players in the same group has a certain influence on the evolution of the game,both the incentive and the inhibition.Finally,the accuracy and validity of the model and method are verified by the experimental simulation,which provides a new theoretical support for solving the information security problems in the real.

Key words: strategy dependency, strength coefficient, improving replicator dynamic, replicator dynamic rate, Jacobian matrix, optimal defense strategy

中图分类号:

TP390

黄健明,张恒巍. 基于改进复制动态演化博弈模型的最优防御策略选取[J]. 通信学报, 2018, 39(1): 170-182.

Jianming HUANG,Hengwei ZHANG. Improving replicator dynamic evolutionary game model for selecting optimal defense strategies[J]. Journal on Communications, 2018, 39(1): 170-182.

图/表 23

图1

图2

表1

攻防博弈复制动态系统均衡点对应的雅可比行列式和迹计算式"

均衡点(q₁,p₁)	$J$ 的行列式和迹的计算式
A (0,0)	$\det J = α_{1} β_{1} (d_{21} - λ_{21} d_{22}) (a_{12} - γ_{21} a_{22})$
	$tr J = α_{1} (d_{21} - λ_{21} d_{22}) + β_{1} (a_{12} - γ_{21} a_{22})$
B (0,1)	$\det J = - α_{1} β_{1} (d_{11} - λ_{21} d_{12}) (a_{12} - γ_{21} a_{22})$
	$tr J = α_{1} (d_{11} - λ_{21} d_{12}) - β_{1} (a_{12} - γ_{21} a_{22})$
C (1,0)	$\det J = - α_{1} β_{1} (d_{21} - λ_{21} d_{22}) (a_{11} - γ_{21} a_{21})$
	$tr J = - α_{1} (d_{21} - λ_{21} d_{22}) + β_{1} (a_{11} - γ_{21} a_{21})$
D (1,1)	$\det J = α_{1} β_{1} (d_{11} - λ_{21} d_{12}) (a_{11} - γ_{21} a_{21})$
	$tr J = - α_{1} (d_{11} - λ_{21} d_{12}) - β_{1} (a_{11} - γ_{21} a_{21})$

表1

表2

攻防博弈演化稳定判别依据（d11?d21?λ21d12+λ21d22=0，a11?a12?γ21a21+γ21a22=0）"

情形	判别依据	A(0,0)	B(0,1)	C(1,0)	D(1,1)
	$\det J$	+	–	–	+
情形1	$tr J$	–	不定	不定	+
	稳定性	ESS	鞍点	鞍点	不稳定
	$\det J$	–	+	+	–
情形2	$tr J$	不定	–	+	不定
	稳定性	鞍点	ESS	不稳定	鞍点
	$\det J$	–	+	+	–
情形3	$tr J$	不定	+	–	不定
	稳定性	鞍点	不稳定	ESS	鞍点
	$\det J$	+	–	–	+
情形4	$tr J$	+	不定	不定	–
	稳定性	不稳定	鞍点	鞍点	ESS

表2

表3

攻防博弈演化稳定判别依据（d11?d21?λ21d12+λ21d22=0，a11?a12?γ21a21+γ21a22≠0）"

情形	判别依据	A(0,0)	B(0,1)	C(1,0)	D(1,1)
	$\det J$	+	–	–	+
情形1	$tr J$	–	不定	不定	+
	稳定性	ESS	鞍点	鞍点	不稳定
	$\det J$	+	–	–	–
情形2	$tr J$	–	不定	+	不定
	稳定性	ESS	鞍点	鞍点	鞍点
	$\det J$	–	+	+	–
情形3	$tr J$	不定	+	–	不定
	稳定性	鞍点	不稳定	ESS	鞍点
	$\det J$	–	+	–	+
情形4	$tr J$	不定	+	不定	–
	稳定性	鞍点	不稳定	鞍点	ESS
	$\det J$	+	+	–	+
情形5	$tr J$	不定	–	不定	+
	稳定性	不定	ESS	鞍点	不稳定
	$\det J$	–	+	+	–
情形6	$tr J$	不定	–	+	不定
	稳定性	鞍点	ESS	不稳定	鞍点
	$\det J$	+	–	+	–
情形7	$tr J$	+	不定	–	不定
	稳定性	不稳定	鞍点	ESS	鞍点
	$\det J$	+	–	–	+
情形8	$tr J$	+	不定	不定	–
	稳定性	不稳定	鞍点	鞍点	ESS

表3

表4

攻防博弈演化稳定判别依据（d11?d21?λ21d12+λ21d22≠0，a11?a12?γ21a21+γ21a22=0）"

情形	判别依据	A(0,0)	B(0,1)	C(1,0)	D(1,1)
	$\det J$	+	–	–	+
情形1	$tr J$	–	不定	不定	+
	稳定性	ESS	鞍点	鞍点	不稳定
	$\det J$	–	+	+	–
情形2	$tr J$	不定	–	+	不定
	稳定性	鞍点	ESS	不稳定	鞍点
	$\det J$	+	+	–	–
情形3	$tr J$	–	+	不定	不定
	稳定性	ESS	不稳定	鞍点	鞍点
	$\det J$	–	–	+	+
情形4	$tr J$	不定	不定	+	–
	稳定性	鞍点	鞍点	不稳定	ESS
	$\det J$	–	–	+	+
情形5	$tr J$	不定	不定	–	+
	稳定性	鞍点	鞍点	ESS	不稳定
	$\det J$	+	+	–	–
情形6	$tr J$	+	–	不定	不定
	稳定性	不稳定	ESS	鞍点	鞍点
	$\det J$	–	+	+	–
情形7	$tr J$	不定	+	–	不定
	稳定性	鞍点	不稳定	ESS	鞍点
	$\det J$	+	–	–	+
情形8	$tr J$	+	不定	不定	–
	稳定性	不稳定	鞍点	鞍点	ESS

表4

图3

图4

图5

图6

图7

图8

图9

图10

图11

表5

图12

表6

表7

图13

图14

图15

图16

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方法	博弈类型	行为理性	复制动态速率准确性	均衡求解	具体应用
文献[13]方法	随机博弈	完全理性	—	简单	策略选取
文献[14]方法	动态博弈	完全理性	—	一般	策略选取
文献[18]方法	演化博弈	不完全理性	低（未考虑同类策略间的影响）	一般	安全防御
文献[19]方法	演化博弈	不完全理性	低（未考虑同类策略间的影响）	简单	安全防御
本文方法	演化博弈	不完全理性	高（综合考虑并定量描述同类策略间的影响）	详细	策略选取

序号	原子攻击动作名称	网络攻击策略
序号	原子攻击动作名称	AS ₁	AS ₂
a₁	安装Web监听	√
a₂	远程攻击	√
a₃	盗取账户		√
a₄	SSH攻击
a₅	网页攻击	√
a₆	数据库监听		√

序号	原子防御动作名称	网络防御策略
序号	原子防御动作名称	DS ₁	DS ₂
d₁	安装数据库补丁		√
d₂	重装监听程序	√	√
d₃	限制数据分组	√
d₄	删除可疑账户
d₅	重启数据服务器	√	√
d₆	主页修复	√